# Completeness relation of spin matrices

I was reading Hugh Osborne's notes on Conformal Field theory and came across a completeness relation which seems easy to prove but I am unable to do it.

$${(s_{\mu\nu})}_{\alpha}^{\beta}{(s^{\mu\nu})}_{\gamma}^{\delta}=\frac{1}{2}({\gamma_{\mu})}^{\beta}_{\alpha}({\gamma^{\mu})}^{\delta}_{\gamma}={\delta_{\alpha}^{\delta}}{\delta_{\gamma}^{\beta}}-\frac{1}{2}{\delta}_{\alpha}^{\beta}{\delta}_{\gamma}^{\delta}$$,

where the spin matrices $$s_{\mu\nu}$$ are defined by

$$s_{\mu\nu}$$=$$\frac{-1}{2}([{\gamma}_{\mu},{\gamma}_{\nu}])$$

It would be highly helpful if someone can just sketch the proof.

• Which of the two equalities do you have trouble proving? Both of them? – M.Jo Jul 3 at 11:06
• First equality is specific for d=3 I suppose and I proved it. But in the second equality I am getting a factor of 4 wrong. – spyk_speigel Jul 10 at 8:46